> For the complete documentation index, see [llms.txt](https://overleaf-pro.ayaka.space/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://overleaf-pro.ayaka.space/latex/zh-cn/shu-xue/02-subscripts-and-superscripts.md). # 下标和上标 ## 引言 上标和下标的使用在涉及指数、索引以及某些特殊运算符的数学表达式中非常常见。本文将解释如何在简单表达式、积分、求和等中编写上标和下标。 定积分是最常见的数学表达式之一，所以让我们看一个例子： ```latex \[ \int\limits_0^1 x^2 + y^2 \ dx \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Cint%5Climits_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$\int\limits\_0^1 x^2 + y^2 \ dx$$ 按照惯例，LaTeX 中的上标和下标分别使用字符 `^` 和 `_` 和；例如，应用于 $$x$$ 和 $$y$$ 上面的代码片段中的。那些字符也可用于数学符号，例如积分符号（`\int`），如上例所示，其中 `_` 用于设置下限，而 `^` 用于设置上限。 命令 `\limits` 会改变积分中上下限的显示方式；如果没有它，上下限会显示在积分符号旁边，而不是上下： ```latex \[ \int_0^1 x^2 + y^2 \ dx \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Cint_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$\int\_0^1 x^2 + y^2 \ dx$$ ## 更多详细示例 符号 `_` 和 `^` 也可以组合在同一个表达式中，例如： ```latex \[ a_1^2 + a_2^2 = a_3^2 \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+2\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+a_1%5E2+%2B+a_2%5E2+%3D+a_3%5E2+%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$a\_1^2 + a\_2^2 = a\_3^2$$ 如果表达式包含较长的上标或下标，这些内容需要用花括号括起来，因为 LaTeX 通常只将数学命令 `^` 和 `_` 应用于后面的单个字符： ```latex \[ x^{2 \alpha} - 1 = y_{ij} + y_{ij} \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+3\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+x%5E%7B2+%5Calpha%7D+-+1+%3D+y_%7Bij%7D+%2B+y_%7Bij%7D++%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$x^{2 \alpha} - 1 = y\_{ij} + y\_{ij}$$ 下标和上标可以以各种方式嵌套和组合。不过，在嵌套下标/上标时，请记住每个命令都必须作用于一个单独的元素；这可以像上面的例子那样是单个字母或数字，也可以是用花括号或方括号括起来的更复杂的数学表达式。例如： ```latex \[ (a^n)^{r+s} = a^{nr+ns} \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+4\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%28a%5En%29%5E%7Br%2Bs%7D+%3D+a%5E%7Bnr%2Bns%7D++%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$(a^n)^{r+s} = a^{nr+ns}$$ ## 使用下标和上标的运算符 某些数学运算符可能需要下标和上标。最常见的情况是积分 `\int` （请查看 [引言](#introduction)）和求和（`\sum`）运算符，它们的上下限正是通过下标和上标排版的。 ```latex \[ \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} \] ``` [在 Overleaf 中打开此示例。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+5\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Cbegin%7Bdocument%7D%0A%5C%5B+%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%0A%3D+%5Cprod_p+%5Cfrac%7B1%7D%7B1+-+p%5E%7B-s%7D%7D+%5C%5D%0A%5Cend%7Bdocument%7D) 这段 LaTeX 代码生成： $$\sum\_{n=1}^{\infty} \frac{1}{n^s} = \prod\_p \frac{1}{1 - p^{-s}}$$ 对于其他需要下标/上标的常用运算符，请查看 [参考指南](#reference-guide). ## 在 Overleaf 中打开所有代码片段 使用下方提供的链接，将上面的所有示例作为一个 Overleaf 项目打开： ```latex 下面是一些下标和上标简单用法的示例： \[ \int\limits_0^1 x^2 + y^2 \ dx \] \vspace{1cm} 在同一个表达式中使用上标和下标 \[ a_1^2 + a_2^2 = a_3^2 \] \vspace{1cm} 更长的下标和上标： \[ x^{2 \alpha} - 1 = y_{ij} + y_{ij} \] \vspace{1cm} 嵌套的下标和上标 \[ (a^n)^{r+s} = a^{nr+ns} \] \vspace{1cm} 带有下标和上标的数学方程示例 \[ \sum_{i=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} \] \vspace{1cm} 平方根的用法 \[ \sqrt[4]{4ac} = \sqrt{4ac}\sqrt{4ac} \] ``` [将上面的所有示例作为一个 Overleaf 项目打开。](https://www.overleaf.com/docs?engine=pdflatex\&snip_name=Superscripts+example+6\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Ctitle%7BSubscripts+and+Superscripts%7D%0A%5Cauthor%7BExamples+from+Overleaf%7D%0A%5Cbegin%7Bdocument%7D%0A%5Cmaketitle%0AHere+are+some+examples+of+simple+usage+of+subscripts+and+superscripts%3A%0A%0A%5C%5B+%5Cint%5Climits_0%5E1+x%5E2+%2B+y%5E2+%5C+dx+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0AUsing+superscript+and+subscripts+in+the+same+expression%0A%0A%5C%5B+a_1%5E2+%2B+a_2%5E2+%3D+a_3%5E2+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ALonger+subscripts+and+superscripts%3A%0A%0A%5C%5B+x%5E%7B2+%5Calpha%7D+-+1+%3D+y_%7Bij%7D+%2B+y_%7Bij%7D++%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ANested+subscripts+and+superscripts%0A%0A%5C%5B+%28a%5En%29%5E%7Br%2Bs%7D+%3D+a%5E%7Bnr%2Bns%7D+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0AExample+of+a+mathematical+equation+with+subscripts+and+superscripts%0A%0A%5C%5B+%5Csum_%7Bi%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7B1%7D%7Bn%5Es%7D+%3D+%5Cprod_p+%5Cfrac%7B1%7D%7B1+-+p%5E%7B-s%7D%7D+%5C%5D%0A%0A%5Cvspace%7B1cm%7D%0A%0ASquared+root+usage%0A%0A%5C%5B+%5Csqrt%5B4%5D%7B4ac%7D+%3D+%5Csqrt%7B4ac%7D%5Csqrt%7B4ac%7D+%5C%5D%0A%5Cend%7Bdocument%7D) ## 参考指南 其他示例和运算符 | LaTeX 标记 | 显示为 | | --------------------- | ------------------------------------------------------------------------------------------------------------------------------------------- | | `a_{n_i}` | a n i {\displaystyle a\_{n\_{i}}} ![{\displaystyle a\_{n\_{i}}}](/files/c60ceb8ae31d67d1005f5d694e21312898980189) | | `\int_{i=1}^n` | ∫ i = 1 n {\displaystyle \int \_{i=1}^{n}} ![{\displaystyle \int \_{i=1}^{n}}](/files/f52ba27f6877f05470d909d30acacb675ebf896a) | | `\sum_{i=1}^{\infty}` | ∑ i = 1 ∞ {\displaystyle \sum \_{i=1}^{\infty }} ![{\displaystyle \sum \_{i=1}^{\infty }}](/files/5e72244e243318aacdf3f3e22f0e91b9bd3a1bb2) | | `\prod_{i=1}^n` | ∏ i = 1 n {\displaystyle \prod \_{i=1}^{n}} ![{\displaystyle \prod \_{i=1}^{n}}](/files/218894b245df7a0ebb4a757c2729bff9b0f50e71) | | `\cup_{i=1}^n` | ∪ i = 1 n {\displaystyle \cup \_{i=1}^{n}} ![{\displaystyle \cup \_{i=1}^{n}}](/files/32c638bf197ca10bf815cdc1be1f6552dda1734f) | | `\cap_{i=1}^n` | ∩ i = 1 n {\displaystyle \cap \_{i=1}^{n}} ![{\displaystyle \cap \_{i=1}^{n}}](/files/41baa876dbb08ff15122a435fca2840319e6a4e4) | | `\oint_{i=1}^n` | ∮ i = 1 n {\displaystyle \oint \_{i=1}^{n}} ![{\displaystyle \oint \_{i=1}^{n}}](/files/3c5e4d637fc560dfcad25bec85d1946bb131d2dc) | | `\coprod_{i=1}^n` | ∐ i = 1 n {\displaystyle \coprod \_{i=1}^{n}} ![{\displaystyle \coprod \_{i=1}^{n}}](/files/e014902e5aae3cb7ef216eea1ccaa31994c7d277) | 还有 `bigcup` 和 `bigcap` 类似于 `cup` 和 `cap` 但这些用于更大的表达式。 ## 进一步阅读 更多信息请参见 * [希腊字母和数学符号列表](/latex/zh-cn/shu-xue/11-list-of-greek-letters-and-math-symbols.md) * [运算符](/latex/zh-cn/shu-xue/07-operators.md) * [积分、求和与极限](/latex/zh-cn/shu-xue/09-integrals-sums-and-limits.md) * [数学字体](/latex/zh-cn/shu-xue/12-mathematical-fonts.md) * [括号](/latex/zh-cn/shu-xue/03-brackets-and-parentheses.md) * [LaTeX 符号大全](http://www.rpi.edu/dept/arc/training/latex/LaTeX_symbols.pdf) （在 **amssymb** 宏包中可用的数学符号完整列表） * [LaTeX2ε 不那么简短的介绍](http://www.ctan.org/tex-archive/info/lshort/) * [使用 amsmath 对齐方程](/latex/zh-cn/shu-xue/06-aligning-equations-with-amsmath.md) --- # Agent Instructions This documentation is published with GitBook. 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